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Popular >

taylor cos(pix),\at 1/2

Solution

taylor

cos (π x), at \frac{1}{2}

Solution

\frac{- sin ( π at \frac{1}{2} ) π}{1 !} (x - at \frac{1}{2}) + \frac{- π ^{2} cos ( π at \frac{1}{2} )}{2 !} (x - at \frac{1}{2})^{2} + \frac{π ^{3} sin ( π at \frac{1}{2} )}{3 !} (x - at \frac{1}{2})^{3} + \frac{π ^{4} cos ( π at \frac{1}{2} )}{4 !} (x - at \frac{1}{2})^{4} + \dots

Solution steps

Apply the Taylor Formula

= cos (π at \frac{1}{2}) + \frac{\frac{d}{d x} ( cos ( π x ) ) ( at \frac{1}{2} )}{1 !} (x - at \frac{1}{2}) + \frac{\frac{d ^{2}}{d x ^{2}} ( cos ( π x ) ) ( at \frac{1}{2} )}{2 !} (x - at \frac{1}{2})^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( cos ( π x ) ) ( at \frac{1}{2} )}{3 !} (x - at \frac{1}{2})^{3} + \dots

Evaluate Derivatives

= cos (π at \frac{1}{2}) + \frac{- sin ( π at \frac{1}{2} ) π}{1 !} (x - at \frac{1}{2}) + \frac{- π ^{2} cos ( π at \frac{1}{2} )}{2 !} (x - at \frac{1}{2})^{2} + \frac{π ^{3} sin ( π at \frac{1}{2} )}{3 !} (x - at \frac{1}{2})^{3} + \frac{π ^{4} cos ( π at \frac{1}{2} )}{4 !} (x - at \frac{1}{2})^{4} + \dots

Refine

= \frac{- sin ( π at \frac{1}{2} ) π}{1 !} (x - at \frac{1}{2}) + \frac{- π ^{2} cos ( π at \frac{1}{2} )}{2 !} (x - at \frac{1}{2})^{2} + \frac{π ^{3} sin ( π at \frac{1}{2} )}{3 !} (x - at \frac{1}{2})^{3} + \frac{π ^{4} cos ( π at \frac{1}{2} )}{4 !} (x - at \frac{1}{2})^{4} + \dots

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Frequently Asked Questions (FAQ)

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